Proposition 7.27. Suppose fn: G + C is continuous, for n > 1, (fn) converges uniformly...
part (c) 7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
(1) Suppose f :(M, d) + (N,0) is not uniformly continuous. Show that there exist an a > 0 and sequences (Xn) and (yn) in M such that d(Ion, yn) < and o(f(xn), f(n)) > € VnE N. (Hint: Negation of the definition of uniform continuity.)
(1) Let {fn} < C[a, b], and let {xn} c [a, b]. Suppose that fn + f uniformly on [a, b] and In + x (as n +00. Show that limn7 fn(2n) = f(x). [3] To a + + ( f or intuico te fan a ant [Dannt u
suppose that f is uniformly continuous on fn(x)=f(x+1/n) converges uniformly to f on 4.3.1. Using Exercise 3.3.22, show that n! k -w (k-1)(n- k)! (1 -2)"-k dz w=k where 0< p<1, and k and n are positive integers such that k < n. 4.3.1. Using Exercise 3.3.22, show that n! k -w (k-1)(n- k)! (1 -2)"-k dz w=k where 0
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a (5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
Let {fn} ⊆ C([0, 1]). Prove that {fn} converges uniformly on [0, 1] if and only if {fn} is equicontinuous on [0, 1] and converges pointwise on [0, 1].
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2) = 1 n²s if 0 2TO 2n-nar if < 0 if < < < 1 Find limno Sofr (x) dx and Slimnfr () dx and use it to show that {fn} does not converge uniformly. Justify your answer.
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
5. Let fn(x) = x"/n on [0, 1]. Show that (fr)nen converges uniformly to a differentiable function on [0, 1], but (f%) does not converge uniformly neN on [0, 1].